I just have a basic one.
I don’t like eMail notifications, because i check my mails only once in a while.
My tab bar is always overcrowded, so single tabs are tiny and a number wouldn’t be much of a use for me, because i can’t see it, but not that i would dislike the idea.
Haven’t played the Alpha yet, so i can’t judge on that little green indicator thing.
If i had to choose one option i would go for the web notifications, i think it’ll draw the most attention.
About notifications, i think the most important one will be about halted productions, it’s mentioned in the description, but not anymore at the short list below, please don’t forget about them.
There could be a notification for colonizing a new planet, too.
And don’t forget about notifications, when shipconstruction is completed.
Also notifications on sold out trade offers would be helpful.
If there will be asteroid mining, notification of depleted ressources could be useful too.
Notifications of low stock, number customizable by user, is an additional idea.